To move your mail accounts, mailboxes and emails to a new computer manually, locate the "Mail" and "Mail Downloads" folders in your user's Library folder, and move them to the same location on your new computer. Also copy your Mail preferences file, com.apple.Mail.plist, located in your user's Library/Preferences folder to the same location on your new computer. That file contains your account information, along with the usual appearance settings, etc.
Those two items are also the ones to back up if you don't have a .Mac account and use its sync feature.
[robg adds: Basic, but I don't think every newcomer to the Mac will know about it, so here it is. Mail does have an importer which works quite well, but this method will create an exact clone of your old setup...]

